Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C9 · Schur multiplier (H2) = C4 |
| direct product | C5×C42⋊C9 |
| unambiguous semidirect product | C42⋊(C5×C9) |
| unambiguous semidirect product | (C42×C5)⋊C9 |
| derived, Sylow | C42 ⋊ C5×C9 |
| Hall | C42×C5 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | C42⋊C3 |
| commutator, hypocenter | C42 | C5×C9 |
| Frattini | C22×C3 | C5×A4 |
| Fitting | C3×C42×C5 | C3 |
| socle, prime radical | C22×C3×C5 | A4 |
| cosocle | C3×C42 | C3×C5 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C3×C5 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5)(6,7,10)(8,11,15)(9,12,14)(13,16,17)(18,19,21,20,22,24,23,25,26);(6,13,9,8)(10,15,14,17)120 conjugacy classes — view the full character table.
| derived subgroup of | C5⋊#63419 |
| maximal subgroup of | C5×#63419, C5⋊#63419, D5×C42⋊C9, … |
| maximal normal subgroup of | C5×#63419, C5⋊#63419, D5×C42⋊C9, … |
| maximal quotient of | C5×#95⋊C9, C2×C5×C42⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 32);